Problem: In a certain hyperbola, the center is at $(-2,0),$ one focus is at $(-2 + \sqrt{34},0),$ and one vertex is at $(-5,0).$  The equation of this hyperbola can be written as
\[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$
The center of the hyperbola is $(h,k) = (-2,0).$  The distance between the center and one vertex is $a = 3,$ and the distance between the center and one focus is $c = \sqrt{34}.$  Then $b^2 = c^2 - a^2 = 34 - 3^2 = 25,$ so $b = 5.$

Therefore, $h + k + a + b = -2 + 0 + 3 + 5 = \boxed{6}.$